Two types of infinity...

It has been a while since i am occupied in some thoughts and a couple of literary works, with the double nature of the infinity understandable by humans.

The two infinites are the following:

1)The infinite seen as an endless collection of smaller parts, giving when added up a finite sum

An example of this is the progression of numbers like 1+1/2+1/4+1/8+...1/n=2. It almost equals 2, but we have added infinite numbers to reach this finite result.

This type of infinity understanding is the basis for Zeno's paradoxes, where one never gets to the end of a finite space or time, since in order to do that he would have to go through infinite parts.

2) The infinite seen as a collection of parts, giving an infinite sum

An example of this is the addition of all the natural numbers (1+2+3+4+....n). It is at the same time an infinite sum, and one giving an infinity as a result.

This type of infinity understanding is what enables us to draw infinite shapes, and realize them as finite. For example any kid can draw a line on a blackboard, because it views it as finite. In reality it is a sum of infinite parts. The circle can also be drawn, while it is at the same time infinite and periodically repeated.

Since i am thinking of concluding a larger literary work with this subject, i felt like asking you if you have any thoughts about this double nature of our understanding of infinity. While infinity is studied in math, i think the question as to how we can have these two antithetical examinations of it is not answered at all.

Looking forward to your views.

repetition can often be quote as infinity
numerals just end up repeating themselves because ones uses the same number all over again
when you think about this
1 2 3 4 5 for example is counting from 1 to 5
but then it is also adding up all those five numbers together 1+2+3+4=5=15
so one could concludes that counting from 1 to 5 is the same as adding them up to get to 15.

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Originally Posted by Kyriakos

It has been a while since i am occupied in some thoughts and a couple of literary works, with the double nature of the infinity understandable by humans.

The two infinites are the following:

1)The infinite seen as an endless collection of smaller parts, giving when added up a finite sum

An example of this is the progression of numbers like 1+1/2+1/4+1/8+...1/n=2. It almost equals 2, but we have added infinite numbers to reach this finite result.

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One doesn't actually add infinitely many numbers to get this result. The sum is rewritten as an algebraic expression where the n appears in the denominator so that if one let the values for n get very large, the value for that part of the expression becomes 0 and so can be ignored. If one cannot rewrite the sum in this manner then the sum diverges to infinity. So the finite result comes from a finite algebraic process.

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Originally Posted by Kyriakos

This type of infinity understanding is the basis for Zeno's paradoxes, where one never gets to the end of a finite space or time, since in order to do that he would have to go through infinite parts.

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I think quantum mechanics removes any problem with Zeno's paradoxes in the physical world. One cannot break either space or time down into smaller and smaller parts.

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Originally Posted by Kyriakos

2) The infinite seen as a collection of parts, giving an infinite sum

An example of this is the addition of all the natural numbers (1+2+3+4+....n). It is at the same time an infinite sum, and one giving an infinity as a result.

This type of infinity understanding is what enables us to draw infinite shapes, and realize them as finite. For example any kid can draw a line on a blackboard, because it views it as finite. In reality it is a sum of infinite parts. The circle can also be drawn, while it is at the same time infinite and periodically repeated.

Since i am thinking of concluding a larger literary work with this subject, i felt like asking you if you have any thoughts about this double nature of our understanding of infinity. While infinity is studied in math, i think the question as to how we can have these two antithetical examinations of it is not answered at all.

Looking forward to your views.

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There is another concept of infinite that Cantor introduced in the 19th century. He constructed various kinds of infinities based on the number of mathematical objects in a set, such as, natural numbers, real numbers, functions over a real domain, etc. However, using these transfinite numbers makes the word "infinity" vague. One would have to ask which infinity are you referring to? These transfinite numbers (aka infinities) are not the same size. Some are bigger than others. For example, the infinity of the natural numbers is smaller than the infinity of the real numbers between 0 and 1.

I don't think I understand the example you provided, but the limit process allowing one to sum an infinite series and the transfinite numbers do result in two different approaches to infinity. The universe does not contain infinitely many quanta, so the transfinite numbers are just a mathematical theory without physical application, although the limit process is used as a technique in calculus to study the physical world.

Be careful not to use limits for ulterior conclusions that must remain infinite. There is no connection between infinite and finite mathematics. Also, an expresssion of infinite mathematics has no connection with finite mathematics. So, when you say "the finite result comes from a finite algebraic process," you are in error. The finite result comes from executing a limit for practical purposes only.

Regarding Zeno, the poor man was insane, so he didn't see how large his foot was.

on second thoughts I am not getting the concept of having two types of infinity since the word itself is singular.
all words are created in a singular fashion and so infinity by this order of logic is one and only singular one way concept.

Infinity is a vague term and for finite, practical purposes it is used as a value that some variable can approach but never actually reach. That is how the calculus viewed it. In that sense there is only one vague infinity which is not a number but a direction, signaled by plus or minus, to approach but never reach.

When Cantor introduced his transfinite numbers, he was talking about a completed infinity like the number of elements in the set of all natural numbers or the number of elements in the set of all real numbers. Such sets don't exist in the physical world, which is finite, but mathematical objects such as integers and real numbers and functions over the real numbers can have infinitely many elements. It turns out that even though the set of natural numbers and real numbers are both infinite, there are more real numbers than natural numbers since they cannot be placed into a one-to-one mapping with each other. That was Cantor's discovery in the 19th century and so now there are infinitely many transfinite (or vaguely, infinite) numbers none of which have any reality in the physical universe which is finite and even had a beginning 13.73 billion years ago confirmed by 21st century exploration.

I don't agree that the physical, so called "universe" is finite. In fact, I call that one the Verse of the UNI, which does not occur in the physical sense.
Regarding exploration, we are looking a light years in the past. We cannot make a single statement about astronomy that can be verified at that level. What's going on today out there might no even resemble in a small percentage our postulations. I'd rather enjoy Italo Calvino's Cosmicomics, or some humorous work on astrology when it comes to a so-called universe that could not occur for anyone with sense. If you are not going to make sense, use something that is fun to play with against the idiotic framework because it makes no sense and thereby is not so cruelly stupid.
Have some fun.

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Originally Posted by cafolini

I don't agree that the physical, so called "universe" is finite.

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Olber's paradox, perhaps originating in the 17th century, almost guarantees that there is some finitude to the universe, otherwise why is the night sky dark? It should be full of light from the infinitely many stars that have had an infinite amount of time to reach us.

http://en.wikipedia.org/wiki/Olbers%27_paradox

Infinite,another word for the unrepresentable. A human personality is infinite.(or at lest some are !)

I keep getting the idea of "infinite" caught up with quantity in my mind, but I suspect human consciousness goes beyond the finite universe in some way.

Regarding Olber's paradox, I think it might also be evidence that there is no multiverse, or that the multiverse, assuming it has some reality, has to somehow be prohibited from breaking into our space and time. Those promoting a multiverse that exists outside our universe would need to show why we haven't seen any light from these other universes.

Yes,infinity is not quantity but quality,AKA human interpersonal awareness.

I like to think of consciousness as beyond space and time out of which the space and time universe of science arose. This makes it non-local (outside space) and eternal (outside time). The space-time universe was created and is supported by it. That's the opposite of those who claim consciousness is a by-product of the finite space-time universe. We are probably saying something similar, but who knows?

Yes,very similiar. I would just clarify that i dont think that conciousness 'creates' things. In other words we are concious of what is not us,and these are different things/entities.

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Originally Posted by Theunderground

Yes,very similiar. I would just clarify that i dont think that conciousness 'creates' things. In other words we are concious of what is not us,and these are different things/entities.

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Good points. I haven't really thought this through. It seems that consciousness is where the choice comes to create something or not and there are others outside us as well who are also conscious. So if there is some kind of choice, no matter how restricted it may appear, there is some kind of consciousness.

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Originally Posted by Kyriakos

It has been a while since i am occupied in some thoughts and a couple of literary works, with the double nature of the infinity understandable by humans.

The two infinites are the following:

1)The infinite seen as an endless collection of smaller parts, giving when added up a finite sum

2) The infinite seen as a collection of parts, giving an infinite sum

Since i am thinking of concluding a larger literary work with this subject, i felt like asking you if you have any thoughts about this double nature of our understanding of infinity. While infinity is studied in math, i think the question as to how we can have these two antithetical examinations of it is not answered at all.

Looking forward to your views.

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I can totally relate with this. There is a very direct analogy to the world of people.

The first type is analogous to the chaos in one man's mind. The chaos of thoughts, perceptions, instincts, opinions and judgments. You can try to resolve it infinitely but some parts will still remain tangled. You can try to put words to the chaos, try to come up with a theory, an analysis or a model but some parts will still remain unaddressed. And yet this chaos is limited in that it is limited to one person alone. A person, every person carries this chaos within oneself and every human being is a limited self (the realized self). All these add up to 1 (person) but it takes a leap of imagination and ignorance towards finer aspects of one's personal traits.

The other type of infinity is the world of infinite people. One can argue about the number of people being limited but if one considers interactions between all the people of the world in all possible ways, though mathematically it will still remain a finite number, for all practical purposes let me call it infinite. After all when we say that the physical size of the universe is infinite, the construction of language here is paradoxical in that the concept of size exists only for finite things. I am not interested in addressing this paradox here.
This infinity is far from the reach of all social sciences - philosophy, anthropology, psychology, sociology, political science etc just like the universe is beyond the reach of physics and biology. I have tremendous respect for endeavors in sciences and social sciences and I still have to learn to look beyond the arrogance of people saying that their theory has tamed the universe.

Can the analogy go into finer aspects now? Can we relate the dynamic nature of the people world with something in mathematics? I think a greater study of challenges in the field of mathematics, ones solved in the past and ones that are still a challenge, and in the field of human sciences and literature will be required to address this. But that's not my job, right? That's your job. :D Why should I go on the wikipedia and try to read Number Theory? You go and read it. And do tell me if you find something relevant.

I admire the project. I'd like it to be more dedicated to bringing people in touch with their inner chaos and allowing a bit of chaos in their own lives and perceptions rather than a clinical study and mapping of the two things.

Good luck!

P.S.: I am sorry for people having taken your analogy a bit too literally. I go by the post-structuralist attitude - the truth of a theory/narrative depends upon how convincing it is. Ultimately everything is a narrative. One has the freedom to believe whatever one chooses to believe. To force one's beliefs as objective truths down other people's throats because of technical/linguistic definitions, I do not condone it.